Light propagation in finite and infinite photonic crystals: The recursive Green’s function technique

Abstract

We report a new computational method based on the recursive Green's function technique for calculation of light propagation in photonic crystal structures. The advantage of this method in comparison to the conventional finite-difference time domain (FDTD) technique is that it computes Green's function of the photonic structure recursively by adding slice by slice on the basis of Dyson's equation. This eliminates the need for storage of the wave function in the whole structure, which obviously strongly relaxes the memory requirements and enhances the computational speed. The second advantage of this method is that it can easily account for the infinite extension of the structure both into air and into the space occupied by the photonic crystal by making use of the so-called "surface Green's functions". This eliminates the spurious solutions (often present in the conventional FDTD methods) related to e.g. waves reflected from the boundaries defining the computational domain. The developed method has been applied to study scattering and propagation of the electromagnetic waves in the photonic band-gap structures including cavities and waveguides. A particular attention has been paid to surface modes residing on a termination of a semi-infinite photonic crystal. We demonstrate that coupling of the surface states with incoming radiation may result in enhanced intensity of an electromagnetic field on the surface and very high Q factor of the surface state. This effect can be employed as an operational principle for surface-mode lasers and sensors.

Light propagation in nanorod arrays

“…Namely, it does not account for the shape and distribution of metal clusters in the dielectric medium, neglecting important polarization properties of both single non-circular particles and their arrangements 19,20 . In order to incorporate these features and study transmission characteristics of periodic and disordered nanorod arrays we apply the recursive Green's function technique 21 .…”

Resonant splitting in periodic T-shaped photonic waveguides

“…The periodic T-shaped waveguide we considered is as depicted in Fig 30,31 can be used to calculate the photonic transport properties of the waveguide structure. In terms of the Green's function scheme, the system is divided into a set of effective square lattices with lattice constant  .…”

Frequency comparison of light transmission in a defected quasi-one-dimensional photonic crystal slab

“…This formalism is general, and for TE modes the same procedures have to be reviewed. Extending the method implemented in [12], we discretize Equation 1 since for TM modes the main components of fields are (E z , H x , H y ); therefore, the electric field and the other two Figure 1 A PC slab. A PC slab consisting of high-dielectric rods between low-dielectric background which is sandwiched in two high-dielectric-constant layers parallel to the x-axis direction, (a) the perfect system and (b) the imperfect system, while the central rod is removed.…”

“…The ratio of incident modes of defected systems with respect to clean systems can be calculated from the difference of these individual modes by applying the appropriate photonic Green function. The organization of the paper is as follows: In the 'Methods' section, we extended the method implemented in [12] to discretize Maxwell's equations of our system. One of these equations is used for transverse magnetic (TM) modes to introduce an eigenvalue-eigenfunction equation of the PC slab in tight binding method.…”

“…In recent experimental research, more attention has been given to photon localization in PC microcavities and wave guides [3][4][5], nanocavities [6], and disordered or amorphous PCs [7][8][9]. Theoretical investigations have mostly used perturbation theory completed with Green tensor [10,11], tight binding method [12][13][14], Wannier function [15], and Lippmann-Schwinger formalism [16]. Well-engineered designs and proper materials lead to less refraction at the boundaries; therefore, light transports through the bulk by interference in the internal structure while the occurrence of boundaries affects the dispersion relation as reported for metallic PC resonators [17] and also limited boundaries with antireflection structures [18].…”